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Multilinear estimates for periodic KdV equations, and applications. (English) Zbl 1062.35109
The authors deal with the Cauchy problem for periodic generalized Korteweg-de Vries (KdV) equations of the form $\begin{cases} \partial_t u+\frac{1}{4\pi^2} \partial^3_x u+ (F(u))_x= 0,\quad & u:\mathbb{T}\times [0,T]\to \mathbb{R},\\ u(x,0)= u_0(x),\quad & x\in\mathbb{T},\end{cases}\tag{1}$ where $$F$$ is a polynomial of degree $$k+ 1$$, the initial data $$u_0$$ is in a Sobolev space $$H^s(\mathbb{T})$$, and $$\mathbb{T}= \mathbb{R}/\mathbb{Z}$$ is the torus. The main result is a sharp multilinear estimate which allows the authors to prove that (1) is locally well-posed in $$H^s(\mathbb{T})$$ for $$s> 1/2$$.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 42B35 Function spaces arising in harmonic analysis 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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